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Our goal in Bayesian modeling is, at least largely, to find the most accurate representation of a real system about which we may be receiving inconsistent expert advice, rather than finding ways of modeling the inconsistency itself. Bayesian networks provide a natural representation of probabilities which allow for (and take advantage of) any independencies that may hold, while not being limited to problems satisfying strong independence requirements.

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The combination of substantial increases in computer power with the Bayesian networks ability to use any existing independencies to computational advantage make the approximations and restrictive assumptions of earlier uncertainty formalisms pointless. So we now turn to the main game: understanding and representing uncertainty with probabilities.

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After applying Bayes theorem to obtain P(h|e) adopt that as your posterior degree of belief in h Conditionalization, in other words, advocates belief updating via probabilities conditional upon the available evidence. It identifies posterior probability (the probability function after incorporating the evidence, which we are writing Bel(.) ) with conditional probability

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Generally, agents are able to assign utility (or, value) to the situations in which they find themselves. We know what we like, we know what we dislike, and we also know when we are experiencing neither of these. Given a general ability to order situations, and bets with definite probabilities of yielding particular situations, Frank Ramsey [231] demonstrated that we can identify particular utilities with each possible situation, yielding a utility function

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If we have a utility function U(O i |A) over every possible outcome of a particular action A we are contemplating, and if we have a probability for each such outcome P(O i |A) then we can compute the probability-weighted average utility for that action otherwise known as the expected utility of the action:

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It is commonly taken as axiomatic by Bayesians that agents ought to maximize their expected utility. That is, when contemplating a number of alternative actions, agents ought to decide to take that action which has the maximum expected utility. Utilities have behavioral consequences essentially: any agent who consistently ignores the putative utility of an action or situation arguably does not have that utility.

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Fair bets are fair because their expected utility is zero. Suppose we are contemplating taking the fair bet B on proposition h for which we assign probability P(h). Then the expected utility of the bet is: Sí

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Given that the bet has zero expected utility, the agent should be no more inclined to take the bet in favor of h than to take the opposite bet against h

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Bayesian networks (BNs) are graphical models for reasoning under uncertainty, where the nodes represent variables (discrete or continuous) and arcs represent direct connections between them. These direct connections are often causal connections. In addition, BNs model the quantitative strength of the connections between variables, allowing probabilistic beliefs about them to be updated automatically as new information becomes available.

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A Bayesian network is a graphical structure that allows us to represent and reason about an uncertain domain. The nodes in a Bayesian network represent a set of random variables from the domain. A set of directed arcs (or links) connects pairs of nodes, representing the direct dependencies between variables. Assuming discrete variables, the strength of the relationship between variables is quantified by conditional probability distributions associated with each node. The only constraint on the arcs allowed in a BN is that there must not be any directed cycles There are a number of steps that a knowledge engineer must undertake when building a Bayesian network:

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Example problem: Lung cancer. A patient has been suffering from shortness of breath (called dyspnoea) and visits the doctor, worried that he has lung cancer. The doctor knows that other diseases, such as tuberculosis and bronchitis, are possible causes, as well as lung cancer. She also knows that other relevant information includes whether or not the patient is a smoker (increasing the chances of cancer and bronchitis) and what sort of air pollution he has been exposed to. A positive X-ray would indicate either TB or lung cancer

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First, the knowledge engineer must identify the variables of interest. what are the nodes to represent and what values can they take? The values should be both mutually exclusive and exhaustive, which means that the variable must take on exactly one of these values at a time. Common types of discrete nodes include: Boolean nodes, which represent propositions, taking the binary values true (T) or false (F). E.g. the node Cancer

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Ordered values. For example, a node Pollution might represent a patients pollution exposure and take the values {low, medium, high}. Integral values. For example, a node called Age might represent a patients age and have possible values from 1 to 120. The trick is to choose values that represent the domain efficiently, but with enough detail to perform the reasoning required.

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The structure, or topology, of the network should capture qualitative relationships between variables. In particular, two nodes should be connected directly if one affects or causes the other, with the arc indicating the direction of the effect.

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a node is a parent of a child, if there is an arc from the former to the latter if there is a directed chain of nodes, one node is an ancestor of another if it appears earlier in the chain, whereas a node is a descendant of another node if it comes later in the chain any node without parents is called a root node, while any node without children is called a leaf node. Any other node (non-leaf and non-root) is called an intermediate node. root nodes represent original causes, while leaf nodesrepresent final effects

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Markov blanket

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To specify the probability distribution of a Bayesian network, one must provide two types of probabilities: (1) the prior probabilities of the root nodes (nodes with no parents) and (2) the conditional probabilities of all nonroot nodes given all combinations of its parent nodes. The next step is to quantify the relationships between connected nodes (specifying a conditional probability distribution for each node). Considering discrete variables this takes the form of a conditional probability table (CPT)

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First, for each node we need to look at all the possible combinations of values of those parent nodes. Each such combination is called an instantiation of the parent set. For each distinct instantiation of parent node values, we need to specify the probability that the child will take each of its values.

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there are no direct dependencies in the system being modeled which are not already explicitly shown via arcs Bayesian networks which have the Markov property are also called Independence-maps (or, I-maps for short), since every independence suggested by the lack of an arc is real in the system. every arc in a BN happens to correspond to a direct dependence in the system, then the BN is said to be a Dependence-map (or, D-map for short). A BN which is both an I-map and a D-map is said to be a perfect map.

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The process of conditioning (also called probability propagation or inference or belief updating) is performed via a flow of information through the network In our probabilistic system, this becomes the task of computing the posterior probability distribution for a set of query nodes, given values for some evidence (or observation) nodes

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diagnostic reasoning, i.e., reasoning from symptoms to cause predictive reasoning, reasoning from new information about causes to new beliefs about effects, following the directions of the network arcs intercausal reasoning reasoning about the mutual causes of a common effect

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BNs are considered to be representations of joint probability distributions There is a fundamental assumption that there is a useful underlying structure to the problem being modeled that can be captured with a BN A BN gives a more compact representation than simply describing the probability of every joint instantiation of all variables

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Consider a BN containing the n nodes X1 to Xn. A particular value in the joint distribution is represented by Recalling the structure of a BN implies that the value of a particular node is conditional only on the values of its parent nodes, this reduces to

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The condition that allows us to construct a network from a given ordering of nodes using Pearls network construction algorithm Furthermore, the resultant network will be a unique minimal I-map, assuming the probability distribution is positive. The construction algorithm simply processes each node in order, adding it to the existing network and adding arcs from a minimal set of parents such that the parent set renders the current node conditionally independent of every other node preceding it

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It is desirable to build the most compact BN possible, for three reasons. First, the more compact the model, the more tractable it is. It will have fewer probability values requiring specification; it will occupy less computer memory; probability updates will be more computationally efficient. Second, overly dense networks fail to represent independencies explicitly. Third, overly dense networks fail to represent the causal dependencies in the domain.

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The optimal order is to add the root causes first, then the variable(s) they influence directly, and continue until leaves are reached. To understand why, we need to consider the relation between probabilistic and causal dependence.

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The structure of a causal diagram can be used to propagate probabilities in a Bayesian network. One distinguishing feature of Bayesian networks is the ability to reason about problem structure and propagate probabilities accordingly. In this respect, Bayesian networks are really a form of model-based reasoning, a form of inferential reasoning in which problem solving is aided by knowledge of system structure because system behavior is determined from structure.

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Markov property: there are no direct dependencies in the system being modeled which are not already explicitly shown via arcs. d-separation A set of nodes E d-separates two other sets of nodes X and Y if every path from a node in X to a node in Y is blocked given E If X and Y are d-separated by E, then X and Y are conditionally independent given E

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d-separation (direction-dependent separation) The conditional independence in means that knowing the value of B blocks information about C being relevant to A, and viceversa lack of information about B blocks the relevance of C to A whereas learning about B activates the relation between C and A

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These concepts apply between sets of nodes Given the Markov property, it is possible to determine whether a set of nodes X is independent of another set Y, given a set of evidence nodes E Path (Undirected Path) A path between two sets of nodes X and Y is any sequence of nodes between a member of X and a member of Y such that every adjacent pair of nodes is connected by an arc (regardless of direction) and no node appears in the sequence twice

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Blocked path A path is blocked, given a set of nodes E, if there is a node Z on the path for which at least one of three conditions holds: Z is in E and Z has one arc on the path leading in and one arc out (chain). Z is in E and Z has both path arcs leading out (common cause) Neither Z nor any descendant of Z is in E, and both path arcs lead in to Z (common effect).